This is German artist Kurt Schwitters reading “Ursonate”, his maddening nonsense poem “Fümmsbö-wötääzääÜu”, to London audiences in 1942, soon after his arrival in Britain and his release from the Isle of Man enemy-alien detention camp. A BBC crew sent to record the performance left halfway through, and you can hear why. Though critic Herbert Read valiantly compared the poem to James Joyce’s Finnegan’s Wake, British audiences never really got Schwitters.

This show offers us a fresh chance to understand an individualist who died obscure and marginal in the English Lake District in 1948, yet whose multimedia practice – performance, sound poems, installation, painted sculptures, collages of found materials – heralded cross-disciplinary 21st-century art. Postwar figures from Robert Rauschenberg (“I feel like he made it all just for me”) to Damien Hirst have cited him as an influence. Schwitters would have liked the random, fragmentary light-heartedness of Twitter but he also looked back to Malevich and Mondrian: arranging objects – including, at Millbank, skittles, a spoon, a scrubbing brush – on surfaces built up from newspaper, food labels, metal, plywood. He warmed austere European abstraction with his own playful, even romantic, sensibility. He is an intriguing conduit from modernism to postmodernism.

No one pretends his best work was done in Britain, and Tate’s strongest room is the first – “Schwitters in Germany”, a condensed overview of how his “Merz” aesthetic developed in response to Germany’s collapse after the first world war. In “Yes-what? Picture” (1920), corrugated cardboard diagonals counterpointed by painted and wooden geometric forms radiate across an expressive painterly surface – a collage version of suprematism. “Merz Picture with a Green Ring” (1926) is constructivism lite – a dynamic arrangement of wooden coloured blocks offset by irregular pink forms studded with circles, one painted, another a piece of cork. “Doremifasolasido” (1930) is an elegant pink-brown-gold-blue abstraction of overlapping fragments of theatre and bus tickets, newspaper cuttings, wrapping paper – chance elements placed to suggest musical harmony.

‘Dancer’ (1943)

A painter’s rigorous training, refinement, meticulous control, is evident throughout these compositions where colour, texture, and formal balance provide content. Schwitters, though, was aiming for anarchy. “In the war, things were in terrible turmoil. What I had learnt at the academy was of no use to me ... Everything was broken down and new things had to be made out of the fragments; and this is Merz. It was like a revolution within me,” he explained. “Merz” derived from “Kommerz”, a word printed on a newspaper scrap in an early collage, and it deified junk not capitalism.

“The word ‘Merz’ denotes the combination of all conceivable materials for artistic purposes ... a perambulator wheel, wire-netting, string and cotton wool are factors having equal rights with paint,” according to the artist. Occasionally these elements had political meaning – “Merz Picture 29a, Picture with Turning Wheel” (sadly not in this show) is an assemblage of wheels that only turn clockwise, alluding to Germany’s rightward shift. But what dominates is a haphazard, uncertain quality – a mood at once festive, unrestrained, yet weighted by a sense of world instability.

Announcing himself as “citizen and idiot”, Schwitters played out this push-pull of decorum and Dadaist mockery across Europe through the 1920s and 1930s in performances and collages. An extrovert who also sought solitude, he would, after frenetic travels, retreat to create the work he considered his masterpiece: “Merzbau” is a walk-in sculptural installation of pristine, crazily jagged sculpted forms with which he took over room after room of his parents’ spacious Hanover apartment. Allied bombs destroyed “Merzbau”, now only known from photographs; Tate includes fragments from Schwitters’ attempt to rebuild it in Britain months before his death.

Included in Hitler’s Degenerate Art exhibition of 1937, Schwitters emigrated to Norway, then fled on the icebreaker Fridtjof Nansen to Scotland in 1940. His romantic response to nature is evident in “Mz Oslo Fjord” and “Isbreen under Snow” – the loose expressive brushwork recalls Edvard Munch’s later paintings – and in figurative works created in Britain, such as the lyrical grey-cream “Untitled (Roofs of Houses in Douglas, Isle of Man)”, painted, due to shortage of materials, in wax tempera on linoleum. These were more than casual; Schwitters observed landscape and natural forms as a way of replenishing his imagination for his abstract collages and for small, handheld plaster and wood sculptures – “Speed”, “Dancer”, “Two Forms in Rhythm” – made while on the move. Their simple grace brings to mind Cy Twombly’s work in this medium.

The lyricism of such pieces is the paradox of the second part of this show. Schwitters had left behind his wife, works, reputation, colleagues; among touching stories recounted here is his halting correspondence – in English, because both artists now refused to use German – with his Dada friend Raoul Hausmann, who had quit Berlin for Limoges. Yet in isolation in Britain, his work becomes brighter, freer, less resonant with European avant-garde formalism and more arbitrary, even slipshod, in technique, in favour of increasing concern with content: his own autobiography, and images of consumer culture.

‘Doremifasolasido’ (1930)

“Untitled (Glass Flower)” (1940), a petal-shaped form in broken coloured glass, is surely a reference to the smashed dream of German romanticism and its emblem of the blue flower. “British Made” (1940) collages paper, fabric, cellophane, London bus tickets and a German shipping timetable, all integrated by a painted surface, to recount Schwitters’ journey into exile. “Untitled (Ross with Penny)” (1945-1947) is built up on a checked oval game board; Schwitters and his son played this – the penny-counter they used is one of the collage elements – on the Fridtjof. “47.15 pine trees c26” (1946-1947) conjures a forest from vertically placed strips of photographs, airmail envelopes and card.

The colourful landscape background of “En Morn” (1947) is provided by a Golden Morn peaches label, overlaid with a photograph of a young girl’s head, further food labels – evoking dreams of American postwar plenty, juxtaposed with the text, “These are the things we are fighting for”. 1950s British pop art, Richard Hamilton, Eduardo Paolozzi, are a breath away.

This is a story of resilience, unlikely assimilation, the accidental convergence of man and milieu. “I am now the last artist here – all the others are free,” Schwitters wrote to his wife from the camp on the Isle of Man in 1941. “But all things are equal. If I stay here, then I have plenty to occupy myself. If I am released, then I will enjoy freedom. If I manage to leave for the US, then I will be over there. You carry your own joy with you wherever you go.”